Types of Numbers
There are infinitely many different ways to represent the same fraction. Since we see no reason to abandon our brownie analogy, let's stick with it - half of that pan of brownies can be represented by:
1/2 = 2/4 = 3/6 = 4/8 = ...
Different fractions that represent the same value are called equivalent. See how that word begins - "equi-?" What do you suppose that prefix means? Yes, equine. You've stumbled upon the horse-fraction connection. No, wise guy - it means "equal." Same... equal... got it?
Let's start by discussing how you can tell if two fractions are equivalent, and then we'll examine some specific cases that you may encounter. We say that two fractions and are equivalent if p x s = q x r. For example, to see if is equivalent to , you can just crunch some quick numbers: is 1 x 7 equal to 2 x 5? Since the products are not the same, these fractions represent different rational numbers. On the other hand, we know that 2/3 and 8/12 are equivalent, because 2 x 12 is equal to 3 x 8. Chew on that, and .
Here are some equivalent representations of the number 1:
In other words, if we take all the brownies (three pieces out of three, four pieces out of four, and so on), we get the whole panful all to ourselves. We're going to feel real good about that, too, until about 2 in the morning. Those brownies are some pretty vengeful concoctions.
Here are some equivalent representations of the number 0:
If we have no brownies, it doesn't matter what size the brownies are, because we still don't have any brownies! That thought's enough to make a person cry. No brownies at all? Aw, fudge.
Let's examine a little more closely why and are equivalent fractions. Start with a pan that was cut into 2 brownies, and take one of them.
Why these brownies are brick-like, we'd rather not know. Probably best not to look a gift horse in the mouth.
Anyway, this picture represents the fraction . If we cut each brownie in half, the pan will have twice as many brownies as it did before, so the denominator gets multiplied by two. Each individual brownie will also get cut in half, so we'll have twice as many brownies as we did before; therefore, the numerator also gets multiplied by 2. Anything else you want to multiply by 2 as long as we're at it? We're really in a groove - you sure? Nothing? All right then, let's proceed.
This picture represents the fraction , but the shaded portion is the same size as the shaded portion in the picture representing .
These pictures illustrate the fact that They also illustrate the possibility that we at Shmoop have no idea what an actual brownie looks like. We don't get out much.
Okay, so to summarize: If we start with , multiply the numerator by 2 and multiply the denominator by 2, we get a fraction equivalent to . This same idea works for any fraction and any number n; if we multiply the numerator p and the denominator q each by n, we'll get a fraction equivalent to .
One last, extremely practical example that has nothing to do with brownies. Imagine that your head is a numerator and your torso is a denominator. If you get stung by a bee and swell up to twice your normal size, your head will still retain the same ratio to your torso, correct? If you don't believe us, there's only one way to find out...
Now, two final examples and one non-example of equivalent fractions:
is equivalent to 40/48 because .
is equivalent to because 10 = 2 × 5 and 35 = 7 × 5.
is not equivalent to because to get from to means multiplying the numerator by 3 but the denominator by 2.
Sometimes we want to find an equivalent fraction with a particular denominator in order to perform operations (addition/subtraction) on them. Ideally, you'd like to get as good at manipulating fractions to get them to do what you want them to as you are at manipulating your parents to do the same.
You're getting awfully good at replacing question marks with numbers. How did you get so smart?
Just a couple more and then we'll let you off the hook:
1. Find a fraction with denominator 24 that is equivalent to 5/6.
20/24. Multiply by 4.
2. Find a fraction equivalent to 7/11 with denominator 121
77/121 Multiply by 11.
Wanna confuse your friends? Tell them you're headed to 77-121 to get a Slurpee.